The articles and essays in this blog range from the short to the long. Many of the posts are also introductory (i.e., educational) in nature; though, even when introductory, they still include additional commentary. Older material (dating back mainly to 2005) is being added to this blog over time.

Wednesday, 1 July 2015

Does Logic Have its Own Subject Matter?

What
is the subject matter of logic? This, of course, has been answered in
many ways by many different philosophers. For example:

i)
Logic gives us the structure of thought itself.

ii)
Logic provides the foundations of language.

iii)
Logic gives us the structure of the world in that, for example, it
deals with its possibilities and impossibilities.

It's
not the case, of course, that any of these definitions necessarily
contradict the others. Indeed perhaps logic is accounted for by the
sum of these seemingly distinct definitions.

The
Structure of Thought

It
should be said that many philosophers would say that the structure of
thought mirrors the structure of language. Many others reverse this
duality by saying that the structure of language mirrors the
structure of thought. Whatever the answer is, these statements don't
in themselves tell us what is logical about thought and language. So
let us take thought firstly.

Aristotle
famously offered us what he called the “laws of thought”. These
laws are captured in terms of basic Aristotelian logical terms. Take
the “law of excluded middle”. That is, p or not-p. In
terms of thought, we can say that we can either think that p
is the case or p isn't the case. Take Wittgenstein’s
well-known expression of this 'tautology':

“It
will either rain this afternoon or it won't rain this afternoon.”

Perhaps
this is a bad example because Wittgenstein himself said that the
above “tells us nothing”. However, it's still the case that we
must think that either p will be the case or not-p will
be the case. Aristotle might have said that this shows us that we
cannot think contradictories. Thus thought is structured around the
basic and fundamental fact that we can't think both p and
not-p (at least not at
the same time).

Another
interesting point to make about Aristotelian laws of thought is that
they also apply to the world. More than that, they apply to
everything. So instead of

p
or not-p

we
can have:

A
or not-A

Here
the ‘A’ above can be about any situation in the world. Take the
rain example again. We can at one time say:

We
cannot think, or believe, both that it will and will not rain this
afternoon.

Though
we can say of the world:

“It
can't rain and not rain at one and the same time.”

We
can say here, then, that either the structure of thought mirrors the
structure of language or that the structure of language mirrors the
structure of thought. We can also say that both thought and language
abide by the structure of the world in that neither thought nor
language can break (as it were) the world’s own logical rules.

So
if a ball is painted blue all over it can't also be painted red all
over. That is a de re statement about the world (to use
20th-century jargon). We can also offer de dicto version. We
can't think

“That
ball is red all over.”

at
the same time as thinking

“That
[same] ball is blue all over.”

The
question many philosophers have asked, vis-à-vis thought and
language, is whether or not our thoughts about the ball are
determined by the logical structure of our language. Similarly, do
both language and thought somehow shape how we perceive and
understand the world or does the world somehow impose its structures
on both thought and language?

Let’s
take another Aristotelian law of thought: the law of identity.

This
has something of the blindingly obvious about it. That is, A = A.
However, this law lies at the very heart of all thought, language
and, well, everything. Therefore it deserves to be notated logically
(as Aristotle did) and commented upon.

For
example, if one thought weren't equal to itself (if not numerically
so), then thought itself would cease. Perhaps more relevantly. If we
didn't know that one thought were identical to itself or to another,
this would certainly be the case. In ontological terms, we need to
recognise the identity of a thought-about-an-object over time,
which itself provides us with the very basis of thought itself.
Similarly, A may well equal A; though at one point
people didn't know that the evening star is the same as the morning
star. More relevantly, the names ‘the morning star’ and ‘the
evening star’ have the same reference (to use Frege’s term). That
self-identical reference being the planet Venus.

Without
the stability of the law of identity, thought would be rendered
incoherent, if not void.

The
Structure of Language

Despite
what we said earlier about the mutual relatedness of thought and
language (or “thought and talk” in Donald Davidson’s words), it
can be said that many philosophers have believed that language has it
own “logical grammar”. More relevantly, they've also said that
such grammar is often hidden by the “ordinary grammar” of natural
languages.

Both
Ludwig Wittgenstein and Bertrand Russell (in their early years)
believed that everyday expressions (of a particular type) hide their
logical grammar.

Take
Russell’s “theory of descriptions” in which he uncovered the
logical grammar underneath (if that’s the correct word) statements
such as “The king of France is bald”.

To
repeat: we're asking what is the subject matter of logic. In this
case it's the logical grammar of an everyday expression.

Russell
argued that the English sentence (or statement) “The king of France
is bald” hides it true logical grammar. By careful analysis he
showed that because of the use of the definite article ‘the’ (in
“The king of France is bald”), there is an implicit commitment to
the existence of the king of France. The king of France didn't in
fact exist. Someone who doesn't exist can't be either bald or not
bald. So, to cut Russell’s long and complex story short, this
expression is effectively “meaningless” because of the
description's “empty” definite description.

Other
philosophers have said that the expression’s logical grammar
renders it simply false (for more or less the same reasons). And yet
I myself have used the description “The king of France” in my own
sentences. I've also predicated baldness of this person (or
non-person). Perhaps all I needed was a supply of quotation marks to
escape from what Quine called “Plato’s beard”.

In
retrospect, it can be said that rather than the logical grammar of
that expression being hidden beneath its everyday grammar, its
grammar (both logical and everyday) is explicit and therefore on the
surface; which is more or less what the late Wittgenstein argued. So
instead of offering an analysis, what Russell actually did was offer
us a new version of the given expression. More strongly, it
can be said that he offered us an alternative. That
alternative, however, is free from logical and ontological
complications or ambiguities (or so Russell thought at that time).

Similarly,
the logical positivists’ “verification principle” told us that
certain “metaphysical” expressions are meaningless either because
they can't be verified or because they have no observational or
experiential consequences. The logic of such expressions, therefore,
rendered them “meaningless”(yes, that word again!). In this case,
then, the subject matter of logic is the logical and philosophical
mix-ups of metaphysical and ordinary-language expressions. Only the
expressions of science, maths and logic are free, so the logical
positivists believed, from such sins.

Formal
Logic

More
formally and technically, it can be said that inference, consequence,
validity, entailment, etc. are the true subject matter of logic.
Having said that, as the philosophers and logicians have told us,
even if formal logic does indeed give of the pure and unadulterated
subject matter of logic proper, it's still nevertheless the case that
all language and thought mainly abides by the principles laid down by
formal logic. And even if this isn't always the case, when thought
and language don't do so, this effectively renders them incoherent,
meaningless or useless. In other words, everyone in their everyday
talk infers things, they deduce logical consequences and they believe
in validity and consistency. They just don’t use these technical
terms when they do so.

Of
course we now come across the well-known problem of which way does
the logical arrow point. Does it point from language-users to logic
or from logic to language-users? Do people learn to reason correctly
by a self-conscious use of pre-existing logical laws and rules? Or
does logic simply codify the way people naturally reason and use
logic?

I
suspect that the arrow points in both directions. After all, many
people can improve their logical skills by studying logic in some
form or even acquire brand new logical techniques. Similarly, it's
hard to believe that the majority of people reason correctly by
sitting down and thinking about the laws of logic.

However,
pragmatists like Dewey and C. S. Peirce believed that logicians and
philosophers should codify and notate logical principles and rules
simply by studying the way people reason (or at least scientists) in
their everyday lives. C. S. Peirce believed that it is scientists and
the methods of science that should be the true subject matter of
logic. And Dewey called such bricks-and-mortar logic “everyday
inference”.

So
even if formal logic needn't (or doesn't) study everyday reasoning,
we can safely say that everyday reasoning rarely breaks the rules of
formal logic. This fact isn't unlike Donald Davidson’s argument
that we must assume that the beliefs of the pygmies and our own
beliefs have been, and are, largely true. Similarly, we must also
assume mass rationality rather than mass irrationality. If the latter
were true, then the “radical interpretation” of other communities
(or even aliens from outer space) would be rendered impossible.
Either that or we'd need to assume that such communities were
collectively insane.